Difficult integration using hyperbolic trig functions

I need to calculate the following: , assuming . So far I have tried some things which, quite messily, arrive at a solution but I cannot simplify my result back to the solution in the book, so I think I have taken a more difficult approach.

Here's what I've done:
This can then be solved using a u-substitution, such as , but the result is really ugly, and I don't think this is the path the book would have intended for me to arrive at the simpler result.

I would appreciate a hint. I'm guessing that there's some hyperbolic identity I'm forgetting.